Non-Compact Hopf Maps and Fuzzy Ultra-Hyperboloids

Fuzzy hyperboloids naturally emerge in the geometries of D-branes, twistor theory, and higher spin theories. In this work, we perform a systematic study of higher dimensional fuzzy hyperboloids (ultra-hyperboloids) based on non-compact Hopf maps. Two types of non-compact Hopf maps; split-type and hybrid-type, are introduced from the cousins of division algebras. We construct arbitrary even-dimensional fuzzy ultra-hyperboloids by applying the Schwinger operator formalism and indefinite Clifford algebras. It is shown that fuzzy hyperboloids, $H_F^{2p,2q}$, are represented by the coset, $H_F^{2p,2q}\simeq SO(2p,2q+1)/U(p,q)$, and exhibit two types of generalized dimensional hierarchy; hyperbolic-type (for $q\neq 0$) and hybrid-type (for $q=0$). Fuzzy hyperboloids can be expressed as fibre-bundle of fuzzy fibre over hyperbolic basemanifold. Such bundle structure of fuzzy hyperboloid gives rise to non-compact monopole gauge field. Physical realization of fuzzy hyperboloids is argued in the context of lowest Landau level physics.